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To solve the problem of finding the limit \(\lim_{x \to 0} f(x)\) using the graph provided, let's analyze the behavior of the function \(f(x)\) as \(x\) approaches 0 from both the left and the right.

### Step 1: Examine the graph as \(x\) approaches 0 from the left (\(x \to 0^-\)):
- As \(x\) approaches 0 from the left (values less than 0), the graph indicates that the function \(f(x)\) is approaching the value \(1\).

### Step 2: Examine the graph as \(x\) approaches 0 from the right (\(x \to 0^+\)):
- As \(x\) approaches 0 from the right (values greater than 0), the graph indicates that the function \(f(x)\) is approaching the value \(-2\).

### Step 3: Determine the limit:
- The limit \(\lim_{x \to 0} f(x)\) only exists if the left-hand limit (\(\lim_{x \to 0^-} f(x)\)) and the right-hand limit (\(\lim_{x \to 0^+} f(x)\)) are equal.
- Here, \(\lim_{x \to 0^-} f(x) = 1\) and \(\lim_{x \to 0^+} f(x) = -2\). Since these two limits are not equal, the overall limit \(\lim_{x \to 0} f(x)\) does not exist.

### Answer:
- The correct choice is: **"does not exist because the left- and right-hand limits are not equal."**

Would you like more details on this topic, or do you have any other questions?

Here are five related questions that could further enhance your understanding:
1. How do you evaluate the limit of a piecewise function at a point of discontinuity?
2. What is the significance of left-hand and right-hand limits in determining the overall limit at a point?
3. How can the concept of limits be applied to real-world problems?
4. What are some common types of discontinuities found in graphs of functions?
5. How can a graph help in understanding the behavior of a function near a point of interest?

**Tip:** Always check both the left-hand and right-hand limits when evaluating the limit at a specific point to determine if the overall limit exists.

Use the graph of f shown to find the indicated limit or function value. If the limit does not exist, explain why. What is the value of lim_{x→0} f(x)?

Find the value of the limit of a function using a graph. Understand how to determine whether a limit exists by comparing left-hand and right-hand limits. This example is suitable for students in Grades 11-12.

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